HairBSDF, Mp, upper

Percentage Accurate: 98.5% → 98.8%

Time: 17.6s

Alternatives: 20

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-1 \leq cosTheta\_i \land cosTheta\_i \leq 1\right) \land \left(-1 \leq cosTheta\_O \land cosTheta\_O \leq 1\right)\right) \land \left(-1 \leq sinTheta\_i \land sinTheta\_i \leq 1\right)\right) \land \left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right)\right) \land 0.1 < v\right) \land v \leq 1.5707964\]

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v))
  (* (* (sinh (/ 1.0 v)) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (expf(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinhf((1.0f / v)) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (exp(-((sintheta_i * sintheta_o) / v)) * ((costheta_i * costheta_o) / v)) / ((sinh((1.0e0 / v)) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(exp(Float32(-Float32(Float32(sinTheta_i * sinTheta_O) / v))) * Float32(Float32(cosTheta_i * cosTheta_O) / v)) / Float32(Float32(sinh(Float32(Float32(1.0) / v)) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (exp(-((sinTheta_i * sinTheta_O) / v)) * ((cosTheta_i * cosTheta_O) / v)) / ((sinh((single(1.0) / v)) * single(2.0)) * v);
end

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{-1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{e^{\frac{-1}{v}} \cdot v - \frac{e^{\frac{1}{v}}}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (/ -1.0 v) (* cosTheta_i cosTheta_O))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (- (* (exp (/ -1.0 v)) v) (/ (exp (/ 1.0 v)) (/ 1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((-1.0f / v) * (cosTheta_i * cosTheta_O)) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((expf((-1.0f / v)) * v) - (expf((1.0f / v)) / (1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((((-1.0e0) / v) * (costheta_i * costheta_o)) * exp(((sintheta_o * sintheta_i) / -v))) / ((exp(((-1.0e0) / v)) * v) - (exp((1.0e0 / v)) / (1.0e0 / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(-1.0) / v) * Float32(cosTheta_i * cosTheta_O)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(exp(Float32(Float32(-1.0) / v)) * v) - Float32(exp(Float32(Float32(1.0) / v)) / Float32(Float32(1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((single(-1.0) / v) * (cosTheta_i * cosTheta_O)) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((exp((single(-1.0) / v)) * v) - (exp((single(1.0) / v)) / (single(1.0) / v)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. /-rgt-identity N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{v}{1}}} \]

   3. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{1}{\frac{1}{v}}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \frac{1}{\color{blue}{\frac{1}{v}}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]

   6. lower-/.f32 99.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{\frac{1}{v}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   9. lift-sinh.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\sinh \left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \color{blue}{\left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{v}\right)}{\frac{1}{v}}} \]

   12. distribute-neg-frac N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{v}\right)\right)}}{\frac{1}{v}}} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{v}}\right)\right)}{\frac{1}{v}}} \]

   14. sinh-neg N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\left(\mathsf{neg}\left(\sinh \left(\frac{-1}{v}\right)\right)\right)}}{\frac{1}{v}}} \]

   15. lift-sinh.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh \left(\frac{-1}{v}\right)}\right)\right)}{\frac{1}{v}}} \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\mathsf{neg}\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}}{\frac{1}{v}}} \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\frac{1}{v}}} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\frac{1}{v}}} \]

   19. metadata-eval 99.0

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{-2} \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

6. Applied rewrites 99.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}}} \]

   2. frac-2neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\mathsf{neg}\left(-2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}{\mathsf{neg}\left(\frac{1}{v}\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\mathsf{neg}\left(\color{blue}{-2 \cdot \sinh \left(\frac{-1}{v}\right)}\right)}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   4. distribute-lft-neg-in N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(-2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{2} \cdot \sinh \left(\frac{-1}{v}\right)}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   6. lift-sinh.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\sinh \left(\frac{-1}{v}\right)}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   7. sinh-undef N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{e^{\frac{-1}{v}} - e^{\mathsf{neg}\left(\frac{-1}{v}\right)}}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\color{blue}{\frac{-1}{v}}} - e^{\mathsf{neg}\left(\frac{-1}{v}\right)}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\mathsf{neg}\left(\color{blue}{\frac{-1}{v}}\right)}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   10. distribute-neg-frac N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{v}}}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\frac{\color{blue}{1}}{v}}}{\mathsf{neg}\left(\frac{1}{v}\right)}} \]

   12. neg-mul-1 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\frac{1}{v}}}{\color{blue}{-1 \cdot \frac{1}{v}}}} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\frac{1}{v}}}{-1 \cdot \color{blue}{\frac{1}{v}}}} \]

   14. div-inv N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\frac{1}{v}}}{\color{blue}{\frac{-1}{v}}}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{e^{\frac{-1}{v}} - e^{\frac{1}{v}}}{\color{blue}{\frac{-1}{v}}}} \]

   16. div-sub N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{e^{\frac{-1}{v}}}{\frac{-1}{v}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}}} \]

   17. lower--.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{e^{\frac{-1}{v}}}{\frac{-1}{v}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}}} \]

8. Applied rewrites 99.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{e^{\frac{-1}{v}}}{\frac{-1}{v}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}}} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{e^{\frac{-1}{v}}}{\frac{-1}{v}}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   2. frac-2neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\mathsf{neg}\left(e^{\frac{-1}{v}}\right)}{\mathsf{neg}\left(\frac{-1}{v}\right)}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   3. div-inv N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{-1}{v}\right)}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{-1}{v}}\right)} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{v}}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot \frac{1}{\frac{\color{blue}{1}}{v}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   7. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot \color{blue}{\frac{v}{1}} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   8. /-rgt-identity N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot \color{blue}{v} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(\mathsf{neg}\left(e^{\frac{-1}{v}}\right)\right) \cdot v} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

   10. lower-neg.f32 99.0

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(-e^{\frac{-1}{v}}\right)} \cdot v - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

10. Applied rewrites 99.0%

    \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(-e^{\frac{-1}{v}}\right) \cdot v} - \frac{e^{\frac{1}{v}}}{\frac{-1}{v}}} \]

11. Final simplification 99.0%

    \[\leadsto \frac{\left(\frac{-1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{e^{\frac{-1}{v}} \cdot v - \frac{e^{\frac{1}{v}}}{\frac{1}{v}}} \]
12. Add Preprocessing

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{-1}{v}\right) \cdot -2}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_i cosTheta_O) (/ 1.0 v))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (/ (* (sinh (/ -1.0 v)) -2.0) (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) * (1.0f / v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((sinhf((-1.0f / v)) * -2.0f) / (1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * costheta_o) * (1.0e0 / v)) * exp(((sintheta_o * sintheta_i) / -v))) / ((sinh(((-1.0e0) / v)) * (-2.0e0)) / (1.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(sinh(Float32(Float32(-1.0) / v)) * Float32(-2.0)) / Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) * (single(1.0) / v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((sinh((single(-1.0) / v)) * single(-2.0)) / (single(1.0) / v));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. /-rgt-identity N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{v}{1}}} \]

   3. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{1}{\frac{1}{v}}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \frac{1}{\color{blue}{\frac{1}{v}}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]

   6. lower-/.f32 99.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{\frac{1}{v}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   9. lift-sinh.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\sinh \left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \color{blue}{\left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{v}\right)}{\frac{1}{v}}} \]

   12. distribute-neg-frac N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{v}\right)\right)}}{\frac{1}{v}}} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{v}}\right)\right)}{\frac{1}{v}}} \]

   14. sinh-neg N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\left(\mathsf{neg}\left(\sinh \left(\frac{-1}{v}\right)\right)\right)}}{\frac{1}{v}}} \]

   15. lift-sinh.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh \left(\frac{-1}{v}\right)}\right)\right)}{\frac{1}{v}}} \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\mathsf{neg}\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}}{\frac{1}{v}}} \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\frac{1}{v}}} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\frac{1}{v}}} \]

   19. metadata-eval 99.0

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{-2} \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

6. Applied rewrites 99.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}}} \]

7. Final simplification 99.0%

   \[\leadsto \frac{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{-1}{v}\right) \cdot -2}{\frac{1}{v}}} \]
8. Add Preprocessing

Alternative 3: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(cosTheta\_i \cdot \frac{1}{v}\right) \cdot cosTheta\_O\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{-1}{v}\right) \cdot -2}{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_i (/ 1.0 v)) cosTheta_O)
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (/ (* (sinh (/ -1.0 v)) -2.0) (/ 1.0 v))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * (1.0f / v)) * cosTheta_O) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((sinhf((-1.0f / v)) * -2.0f) / (1.0f / v));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * (1.0e0 / v)) * costheta_o) * exp(((sintheta_o * sintheta_i) / -v))) / ((sinh(((-1.0e0) / v)) * (-2.0e0)) / (1.0e0 / v))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * Float32(Float32(1.0) / v)) * cosTheta_O) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(sinh(Float32(Float32(-1.0) / v)) * Float32(-2.0)) / Float32(Float32(1.0) / v)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * (single(1.0) / v)) * cosTheta_O) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((sinh((single(-1.0) / v)) * single(-2.0)) / (single(1.0) / v));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v}} \]

   2. /-rgt-identity N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{v}{1}}} \]

   3. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \color{blue}{\frac{1}{\frac{1}{v}}}} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot \frac{1}{\color{blue}{\frac{1}{v}}}} \]

   5. un-div-inv N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]

   6. lower-/.f32 99.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{\sinh \left(\frac{1}{v}\right) \cdot 2}{\frac{1}{v}}}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\sinh \left(\frac{1}{v}\right) \cdot 2}}{\frac{1}{v}}} \]

   8. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{2 \cdot \sinh \left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   9. lift-sinh.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\sinh \left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \color{blue}{\left(\frac{1}{v}\right)}}{\frac{1}{v}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{v}\right)}{\frac{1}{v}}} \]

   12. distribute-neg-frac N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{v}\right)\right)}}{\frac{1}{v}}} \]

   13. lift-/.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \sinh \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{v}}\right)\right)}{\frac{1}{v}}} \]

   14. sinh-neg N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \color{blue}{\left(\mathsf{neg}\left(\sinh \left(\frac{-1}{v}\right)\right)\right)}}{\frac{1}{v}}} \]

   15. lift-sinh.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{2 \cdot \left(\mathsf{neg}\left(\color{blue}{\sinh \left(\frac{-1}{v}\right)}\right)\right)}{\frac{1}{v}}} \]

   16. distribute-rgt-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\mathsf{neg}\left(2 \cdot \sinh \left(\frac{-1}{v}\right)\right)}}{\frac{1}{v}}} \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\frac{1}{v}}} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sinh \left(\frac{-1}{v}\right)}}{\frac{1}{v}}} \]

   19. metadata-eval 99.0

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\frac{\color{blue}{-2} \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

6. Applied rewrites 99.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\color{blue}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot cosTheta\_O\right)}}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot cosTheta\_O\right)}}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

   6. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\left(\frac{1}{v} \cdot cosTheta\_i\right)} \cdot cosTheta\_O\right)}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

8. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{1}{v} \cdot cosTheta\_i\right) \cdot cosTheta\_O\right)}}{\frac{-2 \cdot \sinh \left(\frac{-1}{v}\right)}{\frac{1}{v}}} \]

9. Final simplification 98.9%

   \[\leadsto \frac{\left(\left(cosTheta\_i \cdot \frac{1}{v}\right) \cdot cosTheta\_O\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\sinh \left(\frac{-1}{v}\right) \cdot -2}{\frac{1}{v}}} \]
10. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(cosTheta\_i \cdot \frac{1}{v}\right) \cdot cosTheta\_O\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_i (/ 1.0 v)) cosTheta_O)
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (* 2.0 (sinh (/ 1.0 v))) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * (1.0f / v)) * cosTheta_O) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((2.0f * sinhf((1.0f / v))) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * (1.0e0 / v)) * costheta_o) * exp(((sintheta_o * sintheta_i) / -v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * Float32(Float32(1.0) / v)) * cosTheta_O) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * (single(1.0) / v)) * cosTheta_O) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(2.0) * sinh((single(1.0) / v))) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_O \cdot cosTheta\_i}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. frac-2neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(v\right)}{\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{\mathsf{neg}\left(v\right)} \cdot \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(v\right)} \cdot \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. frac-2neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{-1}{v}} \cdot \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{-1}{v}} \cdot \left(\mathsf{neg}\left(cosTheta\_O \cdot cosTheta\_i\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-1}{v} \cdot \left(\mathsf{neg}\left(\color{blue}{cosTheta\_O \cdot cosTheta\_i}\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. *-commutative N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-1}{v} \cdot \left(\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. distribute-lft-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{-1}{v} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. associate-*r* N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{-1}{v} \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)\right) \cdot cosTheta\_O\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{-1}{v} \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)\right) \cdot cosTheta\_O\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\left(\frac{-1}{v} \cdot \left(\mathsf{neg}\left(cosTheta\_i\right)\right)\right)} \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. lower-neg.f32 98.8

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\left(\frac{-1}{v} \cdot \color{blue}{\left(-cosTheta\_i\right)}\right) \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

6. Applied rewrites 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\left(\frac{-1}{v} \cdot \left(-cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

7. Final simplification 98.8%

   \[\leadsto \frac{\left(\left(cosTheta\_i \cdot \frac{1}{v}\right) \cdot cosTheta\_O\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \]
8. Add Preprocessing

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (* (* cosTheta_i cosTheta_O) (/ 1.0 v))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (* 2.0 (sinh (/ 1.0 v))) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) * (1.0f / v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((2.0f * sinhf((1.0f / v))) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * costheta_o) * (1.0e0 / v)) * exp(((sintheta_o * sintheta_i) / -v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(1.0) / v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) * (single(1.0) / v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(2.0) * sinh((single(1.0) / v))) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. clear-num N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{1}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{1}{v}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot cosTheta\_O\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-*.f32 98.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{1}{v} \cdot \color{blue}{\left(cosTheta\_O \cdot cosTheta\_i\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{1}{v} \cdot \left(cosTheta\_O \cdot cosTheta\_i\right)\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.9%

   \[\leadsto \frac{\left(\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{1}{v}\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \]
6. Add Preprocessing

Alternative 6: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* (/ cosTheta_i v) cosTheta_O) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (* 2.0 (sinh (/ 1.0 v))) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i / v) * cosTheta_O) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((2.0f * sinhf((1.0f / v))) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i / v) * costheta_o) * exp(((sintheta_o * sintheta_i) / -v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i / v) * cosTheta_O) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i / v) * cosTheta_O) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(2.0) * sinh((single(1.0) / v))) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{cosTheta\_O \cdot cosTheta\_i}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. associate-/l* N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_O \cdot \frac{cosTheta\_i}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. lower-/.f32 98.8

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{cosTheta\_i}{v}} \cdot cosTheta\_O\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.8%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.8%

   \[\leadsto \frac{\left(\frac{cosTheta\_i}{v} \cdot cosTheta\_O\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \]
6. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (* (/ cosTheta_O v) cosTheta_i) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (* 2.0 (sinh (/ 1.0 v))) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_O / v) * cosTheta_i) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((2.0f * sinhf((1.0f / v))) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_o / v) * costheta_i) * exp(((sintheta_o * sintheta_i) / -v))) / ((2.0e0 * sinh((1.0e0 / v))) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_O / v) * cosTheta_i) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(2.0) * sinh(Float32(Float32(1.0) / v))) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_O / v) * cosTheta_i) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(2.0) * sinh((single(1.0) / v))) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. associate-/l* N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(cosTheta\_i \cdot \frac{cosTheta\_O}{v}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. lower-/.f32 98.7

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\color{blue}{\frac{cosTheta\_O}{v}} \cdot cosTheta\_i\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Final simplification 98.7%

   \[\leadsto \frac{\left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(2 \cdot \sinh \left(\frac{1}{v}\right)\right) \cdot v} \]
6. Add Preprocessing

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{\frac{-1}{v}}{v}}{e^{\frac{-1}{v}} - e^{\frac{1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (*
  (* cosTheta_i cosTheta_O)
  (/ (/ (/ -1.0 v) v) (- (exp (/ -1.0 v)) (exp (/ 1.0 v))))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (cosTheta_i * cosTheta_O) * (((-1.0f / v) / v) / (expf((-1.0f / v)) - expf((1.0f / v))));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (costheta_i * costheta_o) * ((((-1.0e0) / v) / v) / (exp(((-1.0e0) / v)) - exp((1.0e0 / v))))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(cosTheta_i * cosTheta_O) * Float32(Float32(Float32(Float32(-1.0) / v) / v) / Float32(exp(Float32(Float32(-1.0) / v)) - exp(Float32(Float32(1.0) / v)))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (cosTheta_i * cosTheta_O) * (((single(-1.0) / v) / v) / (exp((single(-1.0) / v)) - exp((single(1.0) / v))));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \cdot v} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{e^{\frac{1}{v}}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\color{blue}{\frac{1}{v}}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v} \]

   6. rec-exp N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) \cdot v} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) \cdot v} \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right) \cdot v} \]

   10. lower-/.f32 98.7

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right) \cdot v} \]

5. Applied rewrites 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot v}} \]

7. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\left(\frac{1}{\left(\sinh \left(\frac{1}{v}\right) \cdot v\right) \cdot 2} \cdot \frac{{\left(e^{sinTheta\_O}\right)}^{\left(\frac{-sinTheta\_i}{v}\right)}}{v}\right) \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)} \]

8. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{\frac{1}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]
9. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   2. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{{v}^{2}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   3. unpow2 N/A

      \[\leadsto \frac{\frac{1}{\color{blue}{v \cdot v}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   4. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{v}}{v}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{v}}{v}}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{v}}}{v}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   7. rec-exp N/A

      \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   10. lower--.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{\color{blue}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   11. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{\color{blue}{e^{\frac{1}{v}}} - e^{\frac{-1}{v}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   12. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{e^{\color{blue}{\frac{1}{v}}} - e^{\frac{-1}{v}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   13. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{e^{\frac{1}{v}} - \color{blue}{e^{\frac{-1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

   14. lower-/.f32 98.6

       \[\leadsto \frac{\frac{\frac{1}{v}}{v}}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

10. Applied rewrites 98.6%

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{v}}{v}}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \cdot \left(cosTheta\_i \cdot cosTheta\_O\right) \]

11. Final simplification 98.6%

    \[\leadsto \left(cosTheta\_i \cdot cosTheta\_O\right) \cdot \frac{\frac{\frac{-1}{v}}{v}}{e^{\frac{-1}{v}} - e^{\frac{1}{v}}} \]
12. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (/ cosTheta_i v) v) cosTheta_O)
  (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i / v) / v) * cosTheta_O) / (expf((1.0f / v)) - expf((-1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i / v) / v) * costheta_o) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i / v) / v) * cosTheta_O) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i / v) / v) * cosTheta_O) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{cosTheta\_i}{{v}^{2}} \cdot \frac{cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i}{{v}^{2}} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{cosTheta\_i}{{v}^{2}} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{cosTheta\_i}{{v}^{2}} \cdot cosTheta\_O}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{cosTheta\_i}{\color{blue}{v \cdot v}} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{cosTheta\_i}{v}}{v}} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{cosTheta\_i}{v}}{v}} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   9. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{cosTheta\_i}{v}}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

   10. rec-exp N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}} \]

   11. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

   13. lower--.f32 N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{\color{blue}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

   14. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{\color{blue}{e^{\frac{1}{v}}} - e^{\frac{-1}{v}}} \]

   15. lower-/.f32 N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\color{blue}{\frac{1}{v}}} - e^{\frac{-1}{v}}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v}}} \]

   17. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\color{blue}{\mathsf{neg}\left(\frac{1}{v}\right)}}} \]

   18. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}} \]

   19. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}} \]

   20. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

   21. lower-/.f32 98.4

       \[\leadsto \frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

8. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\frac{\frac{cosTheta\_i}{v}}{v} \cdot cosTheta\_O}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
9. Add Preprocessing

Alternative 10: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ cosTheta_O (* v v)) cosTheta_i)
  (- (exp (/ 1.0 v)) (exp (/ -1.0 v)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((cosTheta_O / (v * v)) * cosTheta_i) / (expf((1.0f / v)) - expf((-1.0f / v)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((costheta_o / (v * v)) * costheta_i) / (exp((1.0e0 / v)) - exp(((-1.0e0) / v)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(cosTheta_O / Float32(v * v)) * cosTheta_i) / Float32(exp(Float32(Float32(1.0) / v)) - exp(Float32(Float32(-1.0) / v))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((cosTheta_O / (v * v)) * cosTheta_i) / (exp((single(1.0) / v)) - exp((single(-1.0) / v)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

   4. lower-*.f32 58.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

8. Applied rewrites 58.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]

9. Taylor expanded in sinTheta_i around 0

   \[\leadsto \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{{v}^{2} \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
10. Step-by-step derivation

    1. times-frac N/A

       \[\leadsto \color{blue}{\frac{cosTheta\_O}{{v}^{2}} \cdot \frac{cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

    2. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

    3. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}}} \]

    4. lower-*.f32 N/A

       \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O}{{v}^{2}} \cdot cosTheta\_i}}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    5. lower-/.f32 N/A

       \[\leadsto \frac{\color{blue}{\frac{cosTheta\_O}{{v}^{2}}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    6. unpow2 N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{\color{blue}{v \cdot v}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    7. lower-*.f32 N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{\color{blue}{v \cdot v}} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}} \]

    8. rec-exp N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}} \]

    9. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}} \]

    10. metadata-eval N/A

        \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}} \]

    11. lower--.f32 N/A

        \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{\color{blue}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]

    12. lower-exp.f32 N/A

        \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{\color{blue}{e^{\frac{1}{v}}} - e^{\frac{-1}{v}}} \]

    13. lower-/.f32 N/A

        \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\color{blue}{\frac{1}{v}}} - e^{\frac{-1}{v}}} \]

    14. lower-exp.f32 N/A

        \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - \color{blue}{e^{\frac{-1}{v}}}} \]

    15. lower-/.f32 98.4

        \[\leadsto \frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}} \]

11. Applied rewrites 98.4%

    \[\leadsto \color{blue}{\frac{\frac{cosTheta\_O}{v \cdot v} \cdot cosTheta\_i}{e^{\frac{1}{v}} - e^{\frac{-1}{v}}}} \]
12. Add Preprocessing

Alternative 11: 70.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot v}{v \cdot v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (/ (* (* cosTheta_i cosTheta_O) v) (* v v))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (*
   (*
    (/
     (-
      (/ (/ (+ 0.16666666666666666 (/ 0.008333333333333333 (* v v))) v) v)
      -1.0)
     v)
    2.0)
   v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((((cosTheta_i * cosTheta_O) * v) / (v * v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / (((((((0.16666666666666666f + (0.008333333333333333f / (v * v))) / v) / v) - -1.0f) / v) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((((costheta_i * costheta_o) * v) / (v * v)) * exp(((sintheta_o * sintheta_i) / -v))) / (((((((0.16666666666666666e0 + (0.008333333333333333e0 / (v * v))) / v) / v) - (-1.0e0)) / v) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * v) / Float32(v * v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(0.008333333333333333) / Float32(v * v))) / v) / v) - Float32(-1.0)) / v) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((((cosTheta_i * cosTheta_O) * v) / (v * v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / (((((((single(0.16666666666666666) + (single(0.008333333333333333) / (v * v))) / v) / v) - single(-1.0)) / v) * single(2.0)) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. frac-2neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v\right)}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - cosTheta\_i \cdot cosTheta\_O}}{\mathsf{neg}\left(v\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-sub N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(v\right)} - \frac{cosTheta\_i \cdot cosTheta\_O}{\mathsf{neg}\left(v\right)}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{0}{\mathsf{neg}\left(v\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{0}{\mathsf{neg}\left(v\right)} - \color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. frac-sub N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot v - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot v - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. mul0-lft N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0} - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. lower--.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. lower-neg.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \color{blue}{\left(-v\right)} \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. lower-neg.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\color{blue}{\left(-cosTheta\_i\right)} \cdot cosTheta\_O\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-neg.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\color{blue}{\left(-v\right)} \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)} \cdot 2\right) \cdot v} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{v}\right)\right)} \cdot 2\right) \cdot v} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}} \cdot 2\right) \cdot v} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\frac{-1 \cdot \frac{\frac{1}{6} + \frac{1}{120} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 1}{\mathsf{neg}\left(v\right)}} \cdot 2\right) \cdot v} \]

7. Applied rewrites 70.4%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\frac{-1 - \frac{\frac{\frac{0.008333333333333333}{v \cdot v} + 0.16666666666666666}{v}}{v}}{-v}} \cdot 2\right) \cdot v} \]

8. Final simplification 70.4%

   \[\leadsto \frac{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot v}{v \cdot v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{\frac{\frac{0.16666666666666666 + \frac{0.008333333333333333}{v \cdot v}}{v}}{v} - -1}{v} \cdot 2\right) \cdot v} \]
9. Add Preprocessing

Alternative 12: 70.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{2 - \frac{\frac{\frac{-0.016666666666666666}{v}}{v} + -0.3333333333333333}{v \cdot v}}{v} \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (*
   (/
    (-
     2.0
     (/ (+ (/ (/ -0.016666666666666666 v) v) -0.3333333333333333) (* v v)))
    v)
   v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / (((2.0f - ((((-0.016666666666666666f / v) / v) + -0.3333333333333333f) / (v * v))) / v) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / (((2.0e0 - (((((-0.016666666666666666e0) / v) / v) + (-0.3333333333333333e0)) / (v * v))) / v) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(2.0) - Float32(Float32(Float32(Float32(Float32(-0.016666666666666666) / v) / v) + Float32(-0.3333333333333333)) / Float32(v * v))) / v) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / (((single(2.0) - ((((single(-0.016666666666666666) / v) / v) + single(-0.3333333333333333)) / (v * v))) / v) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \cdot v} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{e^{\frac{1}{v}}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\color{blue}{\frac{1}{v}}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v} \]

   6. rec-exp N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) \cdot v} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) \cdot v} \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right) \cdot v} \]

   10. lower-/.f32 98.7

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right) \cdot v} \]

5. Applied rewrites 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot v}} \]

6. Taylor expanded in v around -inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}\right) \cdot v} \]
7. Step-by-step derivation

8. Applied rewrites 70.4%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{\frac{\frac{-0.016666666666666666}{v \cdot v} + -0.3333333333333333}{v}}{v} - 2}{-v} \cdot v} \]
9. Step-by-step derivation

10. Applied rewrites 70.4%

    \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{\frac{\frac{1}{\frac{v}{\frac{-0.016666666666666666}{v}}} + -0.3333333333333333}{v}}{v} - 2}{-v} \cdot v} \]

11. Taylor expanded in v around -inf

    \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{3} + \frac{1}{60} \cdot \frac{1}{{v}^{2}}}{{v}^{2}} - 2}{v}\right) \cdot v} \]
12. Step-by-step derivation

13. Applied rewrites 70.4%

    \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{\frac{\frac{-0.016666666666666666}{v}}{v} + -0.3333333333333333}{v \cdot v} - 2}{-v} \cdot v} \]

14. Final simplification 70.4%

    \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{2 - \frac{\frac{\frac{-0.016666666666666666}{v}}{v} + -0.3333333333333333}{v \cdot v}}{v} \cdot v} \]
15. Add Preprocessing

Alternative 13: 63.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot v}{v \cdot v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v} \cdot 2\right) \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (*
   (/ (* (* cosTheta_i cosTheta_O) v) (* v v))
   (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (* (/ (+ 1.0 (/ 0.16666666666666666 (* v v))) v) 2.0) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return ((((cosTheta_i * cosTheta_O) * v) / (v * v)) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((((1.0f + (0.16666666666666666f / (v * v))) / v) * 2.0f) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = ((((costheta_i * costheta_o) * v) / (v * v)) * exp(((sintheta_o * sintheta_i) / -v))) / ((((1.0e0 + (0.16666666666666666e0 / (v * v))) / v) * 2.0e0) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) * v) / Float32(v * v)) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) / Float32(v * v))) / v) * Float32(2.0)) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = ((((cosTheta_i * cosTheta_O) * v) / (v * v)) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((((single(1.0) + (single(0.16666666666666666) / (v * v))) / v) * single(2.0)) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{cosTheta\_i \cdot cosTheta\_O}{v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   2. frac-2neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{\mathsf{neg}\left(v\right)}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   3. neg-sub0 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - cosTheta\_i \cdot cosTheta\_O}}{\mathsf{neg}\left(v\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   4. div-sub N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(v\right)} - \frac{cosTheta\_i \cdot cosTheta\_O}{\mathsf{neg}\left(v\right)}\right)}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{0}{\mathsf{neg}\left(v\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{cosTheta\_i \cdot cosTheta\_O}{v}\right)\right)}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \left(\frac{0}{\mathsf{neg}\left(v\right)} - \color{blue}{\frac{\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)}{v}}\right)}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   7. frac-sub N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot v - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 \cdot v - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   9. mul0-lft N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0} - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   10. lower--.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{\color{blue}{0 - \left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   12. lower-neg.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \color{blue}{\left(-v\right)} \cdot \left(\mathsf{neg}\left(cosTheta\_i \cdot cosTheta\_O\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\mathsf{neg}\left(\color{blue}{cosTheta\_i \cdot cosTheta\_O}\right)\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   15. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(cosTheta\_i\right)\right) \cdot cosTheta\_O\right)}}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   16. lower-neg.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\color{blue}{\left(-cosTheta\_i\right)} \cdot cosTheta\_O\right)}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   17. lower-*.f32 N/A

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\color{blue}{\left(\mathsf{neg}\left(v\right)\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

   18. lower-neg.f32 98.5

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\color{blue}{\left(-v\right)} \cdot v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \color{blue}{\frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]

5. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}} \cdot 2\right) \cdot v} \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\frac{1 + \frac{1}{6} \cdot \frac{1}{{v}^{2}}}{v}} \cdot 2\right) \cdot v} \]

   2. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\color{blue}{\frac{1}{6} \cdot \frac{1}{{v}^{2}} + 1}}{v} \cdot 2\right) \cdot v} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\color{blue}{\frac{\frac{1}{6} \cdot 1}{{v}^{2}}} + 1}{v} \cdot 2\right) \cdot v} \]

   5. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\frac{\color{blue}{\frac{1}{6}}}{{v}^{2}} + 1}{v} \cdot 2\right) \cdot v} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\color{blue}{\frac{\frac{1}{6}}{{v}^{2}}} + 1}{v} \cdot 2\right) \cdot v} \]

   7. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\frac{\frac{1}{6}}{\color{blue}{v \cdot v}} + 1}{v} \cdot 2\right) \cdot v} \]

   8. lower-*.f32 64.0

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\frac{\frac{0.16666666666666666}{\color{blue}{v \cdot v}} + 1}{v} \cdot 2\right) \cdot v} \]

7. Applied rewrites 64.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{0 - \left(-v\right) \cdot \left(\left(-cosTheta\_i\right) \cdot cosTheta\_O\right)}{\left(-v\right) \cdot v}}{\left(\color{blue}{\frac{\frac{0.16666666666666666}{v \cdot v} + 1}{v}} \cdot 2\right) \cdot v} \]

8. Final simplification 64.0%

   \[\leadsto \frac{\frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot v}{v \cdot v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\left(\frac{1 + \frac{0.16666666666666666}{v \cdot v}}{v} \cdot 2\right) \cdot v} \]
9. Add Preprocessing

Alternative 14: 63.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{v} \cdot v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (* (/ (+ (/ 0.3333333333333333 (* v v)) 2.0) v) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((((0.3333333333333333f / (v * v)) + 2.0f) / v) * v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / ((((0.3333333333333333e0 / (v * v)) + 2.0e0) / v) * v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)) / v) * v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((((single(0.3333333333333333) / (v * v)) + single(2.0)) / v) * v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{v \cdot \left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - \frac{1}{e^{\frac{1}{v}}}\right)} \cdot v} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\color{blue}{e^{\frac{1}{v}}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\color{blue}{\frac{1}{v}}} - \frac{1}{e^{\frac{1}{v}}}\right) \cdot v} \]

   6. rec-exp N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) \cdot v} \]

   7. lower-exp.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - \color{blue}{e^{\mathsf{neg}\left(\frac{1}{v}\right)}}\right) \cdot v} \]

   8. distribute-neg-frac N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{\mathsf{neg}\left(1\right)}{v}}}\right) \cdot v} \]

   9. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\frac{\color{blue}{-1}}{v}}\right) \cdot v} \]

   10. lower-/.f32 98.7

       \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(e^{\frac{1}{v}} - e^{\color{blue}{\frac{-1}{v}}}\right) \cdot v} \]

5. Applied rewrites 98.7%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\left(e^{\frac{1}{v}} - e^{\frac{-1}{v}}\right) \cdot v}} \]

6. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}{v} \cdot v} \]
7. Step-by-step derivation

8. Applied rewrites 63.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{v} \cdot v} \]

9. Final simplification 63.9%

   \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{\frac{0.3333333333333333}{v \cdot v} + 2}{v} \cdot v} \]
10. Add Preprocessing

Alternative 15: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/
  (* (/ (* cosTheta_i cosTheta_O) v) (exp (/ (* sinTheta_O sinTheta_i) (- v))))
  (+ (/ 0.3333333333333333 (* v v)) 2.0)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (((cosTheta_i * cosTheta_O) / v) * expf(((sinTheta_O * sinTheta_i) / -v))) / ((0.3333333333333333f / (v * v)) + 2.0f);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (((costheta_i * costheta_o) / v) * exp(((sintheta_o * sintheta_i) / -v))) / ((0.3333333333333333e0 / (v * v)) + 2.0e0)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(Float32(cosTheta_i * cosTheta_O) / v) * exp(Float32(Float32(sinTheta_O * sinTheta_i) / Float32(-v)))) / Float32(Float32(Float32(0.3333333333333333) / Float32(v * v)) + Float32(2.0)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (((cosTheta_i * cosTheta_O) / v) * exp(((sinTheta_O * sinTheta_i) / -v))) / ((single(0.3333333333333333) / (v * v)) + single(2.0));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2 + \frac{1}{3} \cdot \frac{1}{{v}^{2}}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   2. lower-+.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{1}{3} \cdot \frac{1}{{v}^{2}} + 2}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3} \cdot 1}{{v}^{2}}} + 2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\color{blue}{\frac{1}{3}}}{{v}^{2}} + 2} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{\frac{1}{3}}{{v}^{2}}} + 2} \]

   6. unpow2 N/A

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{\frac{1}{3}}{\color{blue}{v \cdot v}} + 2} \]

   7. lower-*.f32 63.9

      \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\frac{0.3333333333333333}{\color{blue}{v \cdot v}} + 2} \]

5. Applied rewrites 63.9%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{\frac{0.3333333333333333}{v \cdot v} + 2}} \]

6. Final simplification 63.9%

   \[\leadsto \frac{\frac{cosTheta\_i \cdot cosTheta\_O}{v} \cdot e^{\frac{sinTheta\_O \cdot sinTheta\_i}{-v}}}{\frac{0.3333333333333333}{v \cdot v} + 2} \]
7. Add Preprocessing

Alternative 16: 58.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{v}{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 1.0 (/ v (* 0.5 (* cosTheta_i cosTheta_O)))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 1.0f / (v / (0.5f * (cosTheta_i * cosTheta_O)));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 1.0e0 / (v / (0.5e0 * (costheta_i * costheta_o)))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(1.0) / Float32(v / Float32(Float32(0.5) * Float32(cosTheta_i * cosTheta_O))))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(1.0) / (v / (single(0.5) * (cosTheta_i * cosTheta_O)));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

   4. lower-*.f32 58.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

8. Applied rewrites 58.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{v}{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}}} \]

11. Final simplification 58.8%

    \[\leadsto \frac{1}{\frac{v}{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}} \]
12. Add Preprocessing

Alternative 17: 58.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ 0.5 (/ v (* cosTheta_i cosTheta_O))))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f / (v / (cosTheta_i * cosTheta_O));
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 / (v / (costheta_i * costheta_o))
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) / Float32(v / Float32(cosTheta_i * cosTheta_O)))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) / (v / (cosTheta_i * cosTheta_O));
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

   4. lower-*.f32 58.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

8. Applied rewrites 58.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.8%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{v}{cosTheta\_i \cdot cosTheta\_O}}} \]
11. Add Preprocessing

Alternative 18: 58.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (/ (* 0.5 (* cosTheta_i cosTheta_O)) v))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return (0.5f * (cosTheta_i * cosTheta_O)) / v;
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = (0.5e0 * (costheta_i * costheta_o)) / v
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(Float32(0.5) * Float32(cosTheta_i * cosTheta_O)) / v)
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = (single(0.5) * (cosTheta_i * cosTheta_O)) / v;
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

   4. lower-*.f32 58.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

8. Applied rewrites 58.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.0%

    \[\leadsto \frac{\left(cosTheta\_i \cdot cosTheta\_O\right) \cdot 0.5}{\color{blue}{v}} \]

11. Final simplification 58.0%

    \[\leadsto \frac{0.5 \cdot \left(cosTheta\_i \cdot cosTheta\_O\right)}{v} \]
12. Add Preprocessing

Alternative 19: 57.9% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (* (/ cosTheta_O v) cosTheta_i)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_O / v) * cosTheta_i);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * ((costheta_o / v) * costheta_i)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(cosTheta_O / v) * cosTheta_i))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_O / v) * cosTheta_i);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

   4. lower-*.f32 58.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

8. Applied rewrites 58.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]
9. Step-by-step derivation

10. Applied rewrites 58.0%

    \[\leadsto \frac{cosTheta\_O}{v} \cdot \color{blue}{\left(cosTheta\_i \cdot 0.5\right)} \]
11. Step-by-step derivation

12. Applied rewrites 58.0%

    \[\leadsto \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \cdot \color{blue}{0.5} \]

13. Final simplification 58.0%

    \[\leadsto 0.5 \cdot \left(\frac{cosTheta\_O}{v} \cdot cosTheta\_i\right) \]
14. Add Preprocessing

Alternative 20: 58.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v} \end{array} \]

FPCore

(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
 :precision binary32
 (* 0.5 (/ (* cosTheta_i cosTheta_O) v)))

C

float code(float cosTheta_i, float cosTheta_O, float sinTheta_i, float sinTheta_O, float v) {
	return 0.5f * ((cosTheta_i * cosTheta_O) / v);
}

Fortran

real(4) function code(costheta_i, costheta_o, sintheta_i, sintheta_o, v)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: costheta_o
    real(4), intent (in) :: sintheta_i
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: v
    code = 0.5e0 * ((costheta_i * costheta_o) / v)
end function

Julia

function code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	return Float32(Float32(0.5) * Float32(Float32(cosTheta_i * cosTheta_O) / v))
end

MATLAB

function tmp = code(cosTheta_i, cosTheta_O, sinTheta_i, sinTheta_O, v)
	tmp = single(0.5) * ((cosTheta_i * cosTheta_O) / v);
end

Derivation

1. Initial program 98.7%

   \[\frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\left(\sinh \left(\frac{1}{v}\right) \cdot 2\right) \cdot v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 58.0%

   \[\leadsto \frac{e^{-\frac{sinTheta\_i \cdot sinTheta\_O}{v}} \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}}{\color{blue}{2}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   2. lower-/.f32 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{cosTheta\_O \cdot cosTheta\_i}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

   4. lower-*.f32 58.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{cosTheta\_i \cdot cosTheta\_O}}{v} \]

8. Applied rewrites 58.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{cosTheta\_i \cdot cosTheta\_O}{v}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(FPCore (cosTheta_i cosTheta_O sinTheta_i sinTheta_O v)
  :name "HairBSDF, Mp, upper"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -1.0 cosTheta_i) (<= cosTheta_i 1.0)) (and (<= -1.0 cosTheta_O) (<= cosTheta_O 1.0))) (and (<= -1.0 sinTheta_i) (<= sinTheta_i 1.0))) (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0))) (< 0.1 v)) (<= v 1.5707964))
  (/ (* (exp (- (/ (* sinTheta_i sinTheta_O) v))) (/ (* cosTheta_i cosTheta_O) v)) (* (* (sinh (/ 1.0 v)) 2.0) v)))